The Whole Number $n$ Is Such That $n \times (n + 30$\] Is A Prime Number. Find The Value Of The Prime Number.Explain Your Answer Clearly.

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Introduction

In mathematics, prime numbers play a crucial role in various mathematical concepts and theorems. A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself. In this article, we will explore the concept of a whole number $n$ such that $n \times (n + 30)$ is a prime number. We will delve into the mathematical reasoning behind this concept and provide a clear explanation of the prime number involved.

Understanding Prime Numbers

Before we dive into the problem, let's briefly discuss prime numbers. Prime numbers are the building blocks of mathematics, and they have unique properties that make them essential in various mathematical operations. Some of the key properties of prime numbers include:

• A prime number is a positive integer greater than 1.
• A prime number has no positive divisors other than 1 and itself.
• The only even prime number is 2, as all other even numbers can be divided by 2.

The Problem: $n \times (n + 30)$ is a Prime Number

Now, let's focus on the problem at hand. We are given that the whole number $n$ is such that $n \times (n + 30)$ is a prime number. To find the value of the prime number, we need to understand the properties of prime numbers and how they relate to the given expression.

Mathematical Reasoning

Let's start by analyzing the expression $n \times (n + 30)$. We can rewrite this expression as $(n + 15)^2 - 225$. This is a difference of squares, which can be factored as $(n + 15 - 15)(n + 15 + 15)$. Simplifying this expression, we get $(n)(n + 30)$.

Now, let's consider the properties of prime numbers. A prime number has no positive divisors other than 1 and itself. Therefore, if $n \times (n + 30)$ is a prime number, then it must be the case that $n$ and $n + 30$ are relatively prime, meaning they have no common factors other than 1.

Finding the Value of the Prime Number

To find the value of the prime number, we need to find the value of $n$ that satisfies the condition that $n \times (n + 30)$ is a prime number. Let's consider the possible values of $n$ and analyze the resulting expression.

If $n = 1$, then $n \times (n + 30) = 1 \times 31 = 31$, which is a prime number.

If $n = 2$, then $n \times (n + 30) = 2 \times 32 = 64$, which is not a prime number.

If $n = 3$, then $n \times (n + 30) = 3 \times 33 = 99$, which is not a prime number.

If $n = 4$, then $n \times (n + 30) = 4 \times 34 = 136$, which is not a prime number.

If $n = 5$, then $n \times (n + 30) = 5 \times 35 = 175$, which is not a prime number.

If $n = 6$, then $n \times (n + 30) = 6 \times 36 = 216$, which is not a prime number.

If $n = 7$, then $n \times (n + 30) = 7 \times 37 = 259$, which is not a prime number.

If $n = 8$, then $n \times (n + 30) = 8 \times 38 = 304$, which is not a prime number.

If $n = 9$, then $n \times (n + 30) = 9 \times 39 = 351$, which is not a prime number.

If $n = 10$, then $n \times (n + 30) = 10 \times 40 = 400$, which is not a prime number.

If $n = 11$, then $n \times (n + 30) = 11 \times 41 = 451$, which is not a prime number.

If $n = 12$, then $n \times (n + 30) = 12 \times 42 = 504$, which is not a prime number.

If $n = 13$, then $n \times (n + 30) = 13 \times 43 = 559$, which is not a prime number.

If $n = 14$, then $n \times (n + 30) = 14 \times 44 = 616$, which is not a prime number.

If $n = 15$, then $n \times (n + 30) = 15 \times 45 = 675$, which is not a prime number.

If $n = 16$, then $n \times (n + 30) = 16 \times 46 = 736$, which is not a prime number.

If $n = 17$, then $n \times (n + 30) = 17 \times 47 = 799$, which is not a prime number.

If $n = 18$, then $n \times (n + 30) = 18 \times 48 = 864$, which is not a prime number.

If $n = 19$, then $n \times (n + 30) = 19 \times 49 = 931$, which is not a prime number.

If $n = 20$, then $n \times (n + 30) = 20 \times 50 = 1000$, which is not a prime number.

If $n = 21$, then $n \times (n + 30) = 21 \times 51 = 1071$, which is not a prime number.

If $n = 22$, then $n \times (n + 30) = 22 \times 52 = 1144$, which is not a prime number.

If $n = 23$, then $n \times (n + 30) = 23 \times 53 = 1219$, which is not a prime number.

If $n = 24$, then $n \times (n + 30) = 24 \times 54 = 1296$, which is not a prime number.

If $n = 25$, then $n \times (n + 30) = 25 \times 55 = 1375$, which is not a prime number.

If $n = 26$, then $n \times (n + 30) = 26 \times 56 = 1456$, which is not a prime number.

If $n = 27$, then $n \times (n + 30) = 27 \times 57 = 1539$, which is not a prime number.

If $n = 28$, then $n \times (n + 30) = 28 \times 58 = 1624$, which is not a prime number.

If $n = 29$, then $n \times (n + 30) = 29 \times 59 = 1711$, which is not a prime number.

If $n = 30$, then $n \times (n + 30) = 30 \times 60 = 1800$, which is not a prime number.

If $n = 31$, then $n \times (n + 30) = 31 \times 61 = 1891$, which is not a prime number.

If $n = 32$, then $n \times (n + 30) = 32 \times 62 = 1984$, which is not a prime number.

If $n = 33$, then $n \times (n + 30) = 33 \times 63 = 2079$, which is not a prime number.

If $n = 34$, then $n \times (n + 30) = 34 \times 64 = 2176$, which is not a prime number.

If $n = 35$, then $n \times (n + 30) = 35 \times 65 = 2275$, which is not a prime number.

If $n = 36$, then $n \times (n + 30) = 36 \times 66 = 2376$, which is not a prime number.

If $n = 37$, then $n \times (n + 30) = 37 \times 67 = 2479$, which is not a prime number.

If $n = 38$, then $n \times (n + 30) = 38 \times 68 = 2584$, which is not a prime number.

If $n = 39$, then $n \times (n + 30) = 39 \times 69 = 2691$, which is not a prime number.

If $n = 40$, then $n \times (n + 30) = 40 \times 70 = 2800$, which is not a prime number.

If $n = 41$, then $n \times (n + 30) = 41 \times 71 = 2911$, which is not a prime number.

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Q&A: Understanding the Whole Number $n$ and Prime Number

In our previous article, we explored the concept of a whole number $n$ such that $n \times (n + 30)$ is a prime number. We analyzed the properties of prime numbers and how they relate to the given expression. In this article, we will provide a Q&A section to help clarify any doubts and provide further understanding of the concept.

Q: What is the definition of a prime number?

A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself.

Q: How do you determine if a number is prime?

To determine if a number is prime, you can use the following steps:

1. Check if the number is less than 2. If it is, then it is not prime.
2. Check if the number is divisible by any of the prime numbers less than its square root. If it is, then it is not prime.
3. If the number passes the above steps, then it is prime.

Q: What is the relationship between the whole number $n$ and the prime number?

The whole number $n$ is such that $n \times (n + 30)$ is a prime number. This means that the product of $n$ and $n + 30$ must be a prime number.

Q: How do you find the value of the prime number?

To find the value of the prime number, you need to find the value of $n$ that satisfies the condition that $n \times (n + 30)$ is a prime number. We can use the following steps:

1. Start with $n = 1$ and check if $n \times (n + 30)$ is a prime number.
2. If it is not a prime number, then increment $n$ by 1 and repeat the process.
3. Continue this process until you find a value of $n$ that satisfies the condition.

Q: What is the significance of the number 30 in the expression $n \times (n + 30)$?

The number 30 is significant in the expression $n \times (n + 30)$ because it is the difference between the two factors. The expression can be rewritten as $(n + 15)^2 - 225$, which is a difference of squares.

Q: How do you determine if two numbers are relatively prime?

Two numbers are relatively prime if their greatest common divisor (GCD) is 1. You can use the following steps to determine if two numbers are relatively prime:

1. Find the GCD of the two numbers using the Euclidean algorithm.
2. If the GCD is 1, then the two numbers are relatively prime.

Q: What is the relationship between the whole number $n$ and the relatively prime numbers?

The whole number $n$ is such that $n$ and $n + 30$ are relatively prime. This means that the GCD of $n$ and $n + 30$ must be 1.

Q: How do you find the value of the prime number using the relatively prime numbers?

To find the value of the prime number using the relatively prime numbers, you need to find the value of $n$ that satisfies the condition that $n$ and $n + 30$ are relatively prime. We can use the following steps:

1. Start with $n = 1$ and check if $n$ and $n + 30$ are relatively prime.
2. If they are not relatively prime, then increment $n$ by 1 and repeat the process.
3. Continue this process until you find a value of $n$ that satisfies the condition.

Conclusion

In this article, we provided a Q&A section to help clarify any doubts and provide further understanding of the concept of the whole number $n$ and prime number. We discussed the definition of a prime number, how to determine if a number is prime, the relationship between the whole number $n$ and the prime number, and how to find the value of the prime number using the relatively prime numbers. We hope that this article has been helpful in understanding the concept of the whole number $n$ and prime number.

Final Answer

The final answer to the problem is $n = 1$ and the prime number is $31$.